No, the government pays them nothing because there is no vital national interest that the government needs to protect. The commercial enterprises with an interest in the region are then free to hire whoever they choose in defense of their legal interests.
"The original function of a letter of marque (or Letter of Reprisal) was to right a private wrong. For example, when a Dutch merchant has his goods stolen in Germany, and he cannot gain satisfaction for his loss through legal or diplomatic means, he can be granted a Letter of Marque by the Dutch government. Such a letter allows him to "capture" a German merchant to compensate him for his loss. Since the early 18th century it was no longer in use as a means to right a private wrong. The function of the letter of Marque had changed. These letters were now used by governments, as an instrument of State, to augment the National Navy. This gave the state a naval force which could attack the commerce of the enemy at no cost to public funds. The ships captured had to be brought before an Admiralty Court and tried to ensure they were a legal prize, and not the property of a neutral state. The privateers acted on a commission recognised under the Law of Nations. One of the principle clauses of a letter of marque is that of specifically naming the country whose vessels can be legally captured. There were heavy penalties if the property of other nations was violated."
http://www.constitution.org/mil/lmr/marq...